JournalspmVol. 70 / No. 1DOI 10.4171/pm/1924

Convergence of a finite difference method for the KdV and modified KdV equations with L2L^2 data

  • Paulo Amorim

    Universidade de Lisboa, Portugal
  • Mário Figueira

    Universidade de Lisboa, Portugal
Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data cover

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Abstract

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in ), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.